- Signal Definition: A signal is defined as a set of information expressed as a function of independent variables, used as input or output in a system.
- Amplitude Scaling: Changes a signal’s strength, either amplifying it (α > 1) or attenuating it (α < 1).
- Addition and Multiplication: Adding and multiplying signals involve summing or multiplying their amplitudes at each time instance.
- Differentiation and Integration: Differentiation gives the rate of change, and integration provides the cumulative sum of continuous signals.
- Operations on Signals: Basic operations include time scaling, reflection, and time shifting, modifying periodicity, mirroring, or moving the signal in time.
A signal is a function that represents information or a measurable quantity. Its independent variable can be time, position or another index. A system can receive signals as inputs and produce signals as outputs.
The following basic signal operations transform signal values or the independent-variable axis. Each definition uses a continuous-time signal x(t); the corresponding discrete-time notation is x(n).
A general transformation can be written as:
Here Y(t) is the transformed signal and X(t) is the original signal. A complete transformation can modify amplitude, time or both.
The basic set of signal operations is organised below by the part of the function that changes.
Basic Signal Operations Performed on Dependent Variables
These operations act on the dependent value at each point in the signal’s domain. They do not replace t or n with a different independent-variable expression.
- Amplitude scaling.
- Pointwise addition.
- Pointwise multiplication.
- Continuous-time differentiation.
- Continuous-time integration.
The definitions below state the domain and assumptions needed for each operation.
Amplitude Scaling of Signals
Amplitude scaling multiplies every signal value by a constant: Y(t) = αX(t). It leaves the time locations of signal features unchanged.
The scalar α controls magnitude and sign:
0<|α|<1 reduces the magnitude.
|α|>1 increases the magnitude. A negative α also reverses the amplitude sign, and α=0 gives the zero signal.
The diagram shows positive scaling by 0.5 in figure (b) and by 1.5 in figure (c).
Addition of Signals
Pointwise addition combines signals defined on a common domain. For every t, the output is y(t)=x1(t)+x2(t). If X1(t) and X2(t) are the two inputs, their sum is:
Multiplication of Signals
Pointwise multiplication combines values at the same independent-variable location. For every t, the output is y(t)=x1(t)x2(t). This differs from convolution, which integrates or sums products across shifted values. For signals X1(t) and X2(t), pointwise multiplication gives:
Differentiation of Signals
For a differentiable continuous-time signal, y(t)=dx(t)/dt gives its instantaneous rate of change. A classical derivative is not defined at a jump; generalized-signal analysis represents a jump derivative with an impulse. Discrete-time sequences instead use difference operations such as x(n)-x(n-1).
The figure illustrates derivatives of example waveforms; interpret square-wave edges using the stated generalized-signal convention.
Integration of Signals
A running integral of a continuous-time signal can be defined as y(t)=∫ from -∞ to t of x(τ)dτ when that improper integral exists. Other integrator definitions can use a finite lower limit and an initial condition. Discrete-time accumulation uses a sum rather than an integral.
The output at time t is the signed area accumulated over the chosen interval.
Basic Signal Operations Performed on Independent Variables
These operations replace the independent variable inside x. They move, reverse or scale the locations of signal features. The set includes:
- Time scaling
- Time reversal
- Time shifting.
The sign and magnitude inside the signal argument determine the result.
Time Scaling of Signals
Time scaling forms y(t)=x(βt). It changes the duration and location of time features while preserving their values. If x is periodic, it also changes the period.
Here X(t) is the original signal and β is the time-scaling factor.
A value of |β| above 1 compresses the signal in time. A value between 0 and 1 expands it. A negative β also reverses time. When β=0, the result is the constant value x(0).
Reflection of Signals
Time reversal reflects a continuous-time signal about t=0 or a discrete-time sequence about n=0. It applies to both signal types.
For continuous time, Y(t)=X(-t). For discrete time, y(n)=x(-n).
A signal is even when X(-t)=X(t).
A signal is odd when X(-t)=-X(t).
The diagram shows the reflection about the vertical axis.
Time Shifting of Signals
Time shifting moves every feature without changing its shape. The form Y(t)=X(t-t0) uses t0 as the shift amount.
Here X(t) is the original signal and t0 is the time shift.
When t0>0, X(t-t0) shifts right and represents a delay of t0.
When t0<0, X(t-t0) shifts left and represents an advance by |t0|. The diagram shows the original signal in figure (a), with a right shift and a left shift in figures (b) and (c).
